Repeat sales index models use actual sales prices and time data points to estimate a market-level home price index. Home price index is one of the most important factors in understanding the housing market trends and activities. The present value of a property may be estimated from the price index using the prior sale information. One repeat sales model proposed by Bailey, Muth, and Nourse (the BMN model) specifies that the change in the logarithm price of a property over a known period of time is equal to the change in the logarithmic price index plus an error term. Another repeat sales model by Case and Shiller (the Case-Shiller model) improves the BMN method. Unlike the BMN model that assumes that the error term is independent, the Case-Shiller model assumes that the variance of the error term is a linear function of the time between sales. Case-Shiller (1987) proposed a three-step procedure to estimate their model. In their first step, an OLS is run by regressing log price difference on a set of time dummies. In the second step, the square of the residual of the first step is regressed on a constant term and a variable for the time interval between sales. In the third step, the predicted dependent variable in the second step is used as weights to re-run the first step regression. As a generalization of the Case-Shiller model, researchers have proposed a quadratic-dispersion model that assumes the variance of the error term is a quadratic function of the time between sales of the property.
The quadratic dispersion index model is also estimated by a three-step procedure similar to the Case-Shiller procedure. The difference is in the second step, where the squared residual from the first step is regressed on a constant term, a transaction time interval and a squared transaction time interval. From an economic perspective, it is expected that the cross-sectional dispersion of residual value is positive and non-decreasing with increasing time interval. However, the application of real world data to the quadratic equation may yield a constant term that is negative, which often results in a prediction of negative dispersion when the time interval is short. In addition, the dispersion of the residual values predicted by the quadratic model is likely to decrease when the interval between sales is long. Since the occurrence of negative dispersion and the decreasing dispersion of residual values with increasing sales intervals run contrary to economic reasoning, the prediction by the quadratic model without a modification will produce unusable and inaccurate estimate of the dispersion.
Moreover, the three-stage method of calculating price indices for properties in a particular region is often carried out using data corresponding to thousands, and sometimes millions, of property sales. For each property, there is data that indicate both the prices and the time interval between sales of the property. These large sets of data inevitably contain some inaccurate values and some data that do not reflect market trends due to factors such as non-arms-length transactions and changes of building attributes. It has been determined that faulty data is most likely to reveal itself as outlying data when compared with the entire data set. The inclusion of outlying data in a data set used to calculate the price indices may undesirably skew the determined price index values. Accordingly, to improve the accuracy of price indices determination, a method is needed to systematically eliminate the data that is most likely to be corrupt from the determination process.
Often there is insufficient data to estimate a complete series of index covering the entire time periods continuously. Obviously, the index value for the time periods for which no data are available can not be estimated. A less Obviously scenario is that a complete series index may not be estimated even when there are data in all the time periods. When the index value for a time period can be estimated, either in a complete or incomplete series, the index value may not necessarily be reliable. For example, an index value for a time period estimated by only a few observations is not a good representation of an MSA-level index for that time period. Accordingly, there is a need for determining the time periods and the associated data that bear on the determination of price index that includes a particular time period. There is also a need for determining a complete and reliable price index for situations when there is insufficient data such that the indices for some time periods either cannot be estimated or are unreliable.